Stress Concentration Factor of Expanded Aluminum Tubes Using Finite Element Modeling

This paper discusses the development of semi-empirical relations for the maximum stress concentration factor (SCF) around circular holes embedded in aluminum tubes under various expansion ratios and mandrel angles. Finite element models were developed to study the expansion of a typical aluminum tube with embedded holes of various sizes. An elastic perfectly-plastic material behaviour was used to describe the structural response of the tubes under expansion. Various hole-diameter-to-tubewall-thickness ratios, tube expansion ratios, and mandrel angles were considered to determine the stress state around the hole at zero and 90 degree locations from which the maximum SCF was determined. Semi-empirical relations for the maximum SCF using the Lagrange interpolation formulation were developed. The developed relations were found to predict the SCFs accurately.


Introduction
Solid expandable tubular technology was introduced to the oil industry in the early nineties with the aim of achieving mono-diameter wells as alternatives to existing telescopic well configurations.This led to tremendous savings of as much as 50% of the original cost of the telescopic well configurations._______________________________________ *Corresponding author's e-mail: aseibi@pi.ac.ae screens has become a normal practice used by field engineers to increase productivity by enlarging the cross-sectional area.Sand control screens, which are generally used in horizontal wells, consist of tubes possessing slots ranging from fine at the outer surface to considerably larger-sized holes at the base plate.
The presence of holes can be problematic and presents a source of stress concentration.This may reveal some weaknesses in the structure, especially when expanded to higher expansion ratios.Chitwood et al. (2005) showed that a perforated 13Cr base tubular fractured after 25% expansion and cracks propagated between the perforations, causing a loss of structural integrity.Pilkey 1997 andToubal et al. 2004 studied the problem of stress concentration around holes in flat plates under axial loading.Similarly, (Van Dyke 1965;Tan 1994) studied pressure vessels and their related components under internal pressure and/or axial loading.Steele et al. 1986 andXue et al. 1991 developed an efficient numerical algorithm to determine stress variations in cylindrical vessels with large openings.Subsequently, Dennis and Palazotto 1990 extended the approach to composite structures with cut-outs.However, according to the knowledge of the authors, stress concentration around holes in expanded tubes has never been studied before.Therefore, the present work, which deals with the estimation of stress concentration around holes in expanded tubes, is the first of its kind.This work is complex in nature and represents a moving boundary problem in terms of mandrel motion within the tube passing by the embedded hole while expanding the tube to a prescribed expansion ratio.The stress concentration may induce crack initiation at the notch root which can result in loss of integrity.For this reason, the tangential stress at zero and 90 degree locations around the hole received great attention through the conduct of a finite-element analysis from which SCFs based on the maximum stress were determined.

Finite Element Modeling
Finite element analysis was performed on solid tubes subjected to various expansion ratios, mandrel angles, and hole-diameter-to-tube-wall thickness ratios.
Tubular expansion occurs when the mandrel moves downward and expands the solid tube to a defined expansion ratio (D o -D i )/D i *100 (Fig. 1). Figure 1a shows the geometrical model of a tube with an inner diameter of D i = 54.6 mm, a wall thickness of t = 6 mm, and an embedded circular hole of variable diameter, d.The figure also shows a conical mandrel with mandrel angle and an outer diameter of D o , which can be varied according to expansion requirements.Points 1 and 2 are located at zero and 90 degree positions around the hole and are the focus of this study, where the variations of the tangential stress at both locations were taken as the mandrel moves downward to expand the tube.This expansion process subjects the tube to a compression state ahead of the mandrel, due to the end conditions.A three-dimensional brick element and an analytic rigid body element were used to model the tube and mandrel, respectively, as shown in Fig. 1b.The mandrel/tube interaction was modeled using a Coulomb friction model where a friction coefficient of 0.1 was used.
Proper boundary conditions representative of lab and field conditions are essential to obtaining accurate results.Special attention must be focused on the region in the vicinity of the hole where stress concentration occurs.The zoomed window section in Fig. 1b shows the finite element mesh of the tube where fine meshes were used in the vicinity of the hole due to the high stress concentration experienced during expansion in this region.An elastic perfectly-plastic material model was used to describe the tube's behaviour.The material properties used for the tube were elastic modulus E = 67 GPa, Poisson's ratio = 0.3, and tensile yield stress y = 150 MPa.Tube expansion in this study was performed by constraining all nodes lying at the bottom side of the tube from moving in any direction, whereas all other nodes were free to move.Since the mandrel was modeled as a rigid body, a vertical downward displacement was given to its reference point, which was located along its centre line at the top face to move the mandrel down.

Case Study
In this study, the friction coefficient was taken as 0.1 for the entire finite element analysis.Three mandrel angles of = 10, 22.5, and 30 o , as well as five expansion ratios of e = 5, 10, 15, 20, and 25% were used.In addition, eight hole-diameter-to-tube-wallthickness ratios were used: d/t = 0.1, 0.2, 0.6, 0.8, 1.0, 1.2, and 1.5.The tangential stress of zero and 90 degrees was taken from the finite element results from which the nominal and peak stress values were computed.The generated data were used to determine the maximum SCF at both locations.Peak values of the tangential stress were used to estimate the SCF defined by Chitwood et al. (2005) as: (1) where max (x)| (0.90) is the maximum stress at zero and 90 degrees, and x denotes the mandrel position along the tube during the expansion process.The stress ave is defined as the average stress of the stress variation before the mandrel reaches the hole.It is worth noting that Eqn.(1) differs from the traditional definition of the stress concentration where the nominal stress is taken into account instead of the average stress used in this study.

Results and Discussions
The finite element results focus on the stress state at zero and 90 degrees at locations around the hole from which the maximum SCF can be determined.Figure 2 shows the variation of the drawing force as a function of the mandrel position for various expansion ratios and mandrel angles.In all cases, the figure shows that the force increases with an increase in expansion ratio and mandrel angle.Figure 2d shows a summary of the average drawing force in terms of expansion ratio for three mandrel angles, indicating that the force increases with an increase in both expansion ratio and mandrel angle.
Typical stress variations of 22 and 11 in terms of mandrel position are shown in Figs. 3 and 4. It is worth noting that the tangential stress at points 1 and 2 are respectively compressive and tensile when the mandrel is far away from the hole, since tube expansion was performed under compression.The two stresses become more compressive as the mandrel approaches the hole and changes sign to become tensile, reaching peak values while expanding the hole.
As the mandrel passes the expanded hole, residual compressive stresses were induced at both locations.This behavior is shown in Fig. 5 where the shape variation of the hole during the three stages of expansion is clearly observed.It can be seen that the hole is under compressive and tensile stress state at points 1 and 2, respectively, when the mandrel is far away from the hole.As the mandrel approaches the hole and starts expanding the region in the vicinity of the hole, the hole becomes oval (Fig. 5b) and the stress state at both locations becomes tensile due to the hole's enlargement.As the mandrel passes the expanded zone of the hole, the hole tends to recover; therefore, the residual stress becomes compressive at both locations.Peak values of 11 and 22 were used to compute the maximum K max(0.90) at points 1 and 2 in terms of a hole-diameter-to-tube-wall-thickness ratio as well as mandrel angle and expansion ratio.to the increase of the nominal stress nom and the decrease of the maximum stress max as of the ratio of d/t increases.
It is worth mentioning that the SCF based on max decreases as the expansion ratio increases from 5 to 25% which is also attributed to the decrease of nominal stress as the expansion ratio increases from 5 to 25%.

Semi-analytical Formulations of Stress Concentration
This section aims to develop expressions for the SCFs as functions of the parameters d/t, , and e used in this study.Table 1 shows expressions for the stress concentration factor K max in terms of hole-diameterto-tube-wall thickness for one set of data.These seventh-order polynomials are valid within the range used in this study (0.1 d/t 0.15).To develop such relations, the Lagrange interpolation method, which is an efficient numerical technique, was used.The advantage of this technique is that the data set points need not be arranged in any particular order as long as they are mutually distinct.The analysis started with a data set consisting of one set of either parameter (d/t, , or e) and their corresponding values of (K tmax ) for a well-determined case.Several cases

Conclusions
The results from three-dimensional elastic perfectly-plastic finite-element analyses were presented to compute the SCFs around circular holes in perforated tubes subjected to a compressive expansion process.Over 120 finite element models were run in order to study the effect of the parameters d/t, , and e on the SCFs.It was found that both SCFs were higher at point 2 (90 o location) indicating that failure might originate at this location.In addition, the results showed that the SCF decreased as the hole-diameterto-tube-thickness ratio and mandrel angle increased.Moreover, generalised semi-empirical relations of both SCFs in terms of d/t, , and e were developed using the Lagrange interpolation method.These expressions can reproduce any of the finite element results with the range of the parametric values undertaken in this study.

Figure 1 .
Figure 1.Geometric and finite element models of the tubular expansion process

Figure 4 .
Figure 4. Tangential stresses versus mandrel position for an expansion ratio of 25%, mandrel angle of 30 o , and when d/t = 0.6 Mandrel Displacement (mm) Mandrel Dislacement (mm) (a) Stress at point 1 (b) Stress at point 2

Figure 5 .
Figure 5. Stress contours and shapes of the hole during the complete expansion process

Figure 6 .Figure 7 .Figure 8 .Figure 9 .
Figure 6.SCF as a function of the hole-diameter-to-tube-thickness ratio for 5% expansion ratio and mandrel angles of 0 and 90 degrees

Table 1 .
SCF for a mandrel angle = 10 o (c) After hole expansion